Is there a name for the kind of statement that suggests an infinite recursion?

It is in a way similar to a paradox such as the one with a statement written on each side of a piece of paper. Side one says "The statement on the other side of this page is true"; Side two says "The statement on the other side of this page is false." But the quotation above is not a paradox, an endless contradiction, but an endless reinforcement. What do we call that?

I'm finding it hard even to come up with adequate tags for this question.

@Andy F: Although I gave it an upvote, I'm not satisfied with Zoot's answer, since I'm already using the term "recursive expression" in the title of this question. I think there must be some other word that expresses this concept, though I can't for the life of me think of what it might be. My brain keeps offering me chiasmus but it's not really that.
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RobustoJan 12 '11 at 16:42

Well it's already described as recursion in the question, but yeah. I'm not sure there's an actual answer to this question.
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NoldorinJan 12 '11 at 16:43

3 Answers
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It is not recursion (and certainly not "infinite recursion") because reading the sentence "A metaphor is like a simile" does not make you repeat/recurse on anything, nor does it invoke a smaller version of the same sentence, or anything like that. It merely so happens that "like a simile" is itself a simile. So this is just self-referential at most (it's not the sentence that refers to itself, but just the part "like a simile"), not recursive.

You may also consider the category of words that describe themselves (like "pentasyllabic"), sometimes called autological (though this is for words, rather than sentences), or (imprecisely) call your sentence self-descriptive, self-similar, or self-exemplifying. ("A metaphor is like a simile" is not an example of a metaphor — the subject of the sentence — but of a simile. But "A simile is like a metaphor" would be an example of a simile.)

Something related is sentences that describe themselves, known as autograms:

This autogram contains five a's, one b, two c's, two d's, thirty-one e's, five f's, five g's, eight h's, twelve i's, one j, one k, two l's, two m's, eighteen n's, sixteen o's, one p, one q, six r's, twenty-seven s's, twenty-one t's, three u's, seven v's, eight w's, three x's, four y's, and one z.

or sentences that describe a part of themselves (from Douglas Hofstadter's Gödel, Escher Bach: An Eternal Golden Braid):

"Is composed of five words" is composed of five words.

(In your sentence it's the other way round: a part of your sentence, but not your sentence itself, describes the whole sentence.)

More from GEB:

This sentence is meaningless because it is self-referential.
This sentence no verb.
This sentence is false. (Epimenides paradox)
The sentence I am now writing is the sentence you are now reading.

The self-reference in most of these is achieved by directly using "this sentence" or equivalent; in your sentence it is achieved more subtly. (So "subtle self-reference" is another phrase that may work for your sentence.) GEB has an entire chapter on self-reference and self-replication; perhaps sentences of your type are mentioned there.

And yet "A metaphor is like a simile" is open-ended, since a simile is a type of metaphor. And self-reference is one definition of recursion. A procedure is recursive if one of the steps that makes up the procedure calls for a new running of the procedure. The example I quoted makes me run the definitions of simile and metaphor over and over again in my head. Am I the only one?
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RobustoJan 12 '11 at 17:50

@Cerberus: Thanks, that's interesting… so if I understand it correctly, declarations are (for our purposes) sentences like "I now pronounce you husband and wife" that become 'true' (or perform their function) by their own utterance?
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ShreevatsaRJan 12 '11 at 18:10

@Robusto: Self-reference is only part of recursion, not one of its definitions. I'm not sure what procedure the sentence "A metaphor is like a simile" specifies (if any), but it doesn't seem to involve a new "running" of the sentence itself. It seems that it must be isomorphic to the procedure that the sentence "A vehicle is like a bus" specifies. And the latter is clearly not recursive. (There is a difference—the latter relies on the meanings of "vehicle" and "bus", and the former on "simile" and "metaphor"—but the meanings don't depend on our sentence itself differently in the two cases.)
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ShreevatsaRJan 12 '11 at 18:12

@ShreevatsaR: Right, that is what it means. We could even broaden the scope of this phenomenon: if I say "you are stupid", I could rephrase that as "I say that you are stupid", which means practically the same; then we could say that this latter statement too was a Declaration, in as much that it became true by being pronounced—the highest-level predicate is "(I) say". // On second thought, this all isn't very relevant to the Question's example; it is merely another kind of self-reference.
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CerberusJan 12 '11 at 18:35

But what you have there as an example isn't circular or recursive as I see it. It's a pretty straightforward statement with no dependencies. A metaphor IS like a simile, simple as that.

Now, if you said "a metaphor is like a simile" and I asked what a simile was and you said "a simile is like a metaphor," then you'd have a circular definition.

But if you're talking about the oddness of using the phrase "like a simile" which is in itself a simile, that's kind of self-referential, but not explicitly so. By that I mean that the self-reference doesn't involve the subject of the sentence. More explicitly self-referential would be where simile is the subject of the sentence and also involved in a simile: "a simile is like..."

+1 for "isn't circular or recursive as I see it. It's a pretty straightforward statement with no dependencies", and "kind of self-referential, but not explicitly so". Your answer came in while I was typing my long answer, and if I had seen it I wouldn't have posted so much verbiage. :-)
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ShreevatsaRJan 12 '11 at 17:29

Sorry, I don't think this is a recursive statement. Recursion and self-reference are related, but not the same. (Or rather, recursion always involves (a form of) self-reference, but not every instance of self-reference involves recursion.)
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ShreevatsaRJan 12 '11 at 17:26

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a recursion is a recursion is a recursion is a recursion is a ...
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EldroßJan 13 '11 at 8:52